A010466 -id:A010466 - cp's OEIS frontend

A378390 Decimal expansion of the surface area of a deltoidal icositetrahedron with unit shorter edge length.

3, 0, 6, 9, 4, 8, 9, 5, 7, 2, 4, 0, 3, 1, 0, 0, 9, 5, 9, 0, 7, 7, 0, 3, 1, 4, 7, 8, 4, 0, 5, 0, 6, 7, 3, 3, 8, 7, 9, 6, 5, 1, 0, 7, 4, 6, 3, 1, 6, 1, 0, 1, 8, 7, 7, 3, 0, 7, 0, 1, 5, 3, 8, 6, 7, 0, 2, 7, 7, 7, 1, 9, 8, 7, 8, 9, 1, 2, 5, 1, 5, 6, 7, 7, 9, 0, 3, 1, 3, 6

Offset: 2

Paolo Xausa, Nov 29 2024

The deltoidal icositetrahedron is the dual polyhedron of the (small) rhombicuboctahedron.

			30.694895724031009590770314784050673387965107463161...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Eric Weisstein's World of Mathematics, Deltoidal Icositetrahedron.
Wikipedia, Deltoidal icositetrahedron.
Index entries for algebraic numbers, degree 4.

Cf. A378391 (volume), A378392 (inradius), A378393 (midradius), A378394 (dihedral angle).
Cf. A343964 (surface area of a (small) rhombicuboctahedron with unit edge).
Cf. A010466.

First[RealDigits[6*Sqrt[29 - Sqrt[8]], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["DeltoidalIcositetrahedron", "SurfaceArea"], 10, 100]]

Equals 6*sqrt(29 - 2*sqrt(2)) = 6*sqrt(29 - A010466).

A041010 Numerators of continued fraction convergents to sqrt(8).

Original entry on oeis.org

2, 3, 14, 17, 82, 99, 478, 577, 2786, 3363, 16238, 19601, 94642, 114243, 551614, 665857, 3215042, 3880899, 18738638, 22619537, 109216786, 131836323, 636562078, 768398401, 3710155682, 4478554083, 21624372014, 26102926097, 126036076402, 152139002499

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..199 [1 removed by _Georg Fischer_, Jul 01 2019]
Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

Cf. A040005 (continued fraction), A041011 (denominators), A010466 (decimals).

Analog for other sqrt(m): A001333 (m=2), A002531 (m=3), A001077 (m=5), A041006 (m=6), A041008 (m=7), A005667 (m=10), A041014 (m=11), A041016 (m=12), ..., A042934 (m=999), A042936 (m=1000).

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[8],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011*)
CoefficientList[Series[(2 + 3*x + 2*x^2 - x^3)/(1 - 6*x^2 + x^4), {x, 0, 30}], x]  (* Vincenzo Librandi, Oct 28 2013 *)
a0[n_] := -((3-2*Sqrt[2])^n*(1+Sqrt[2]))+(-1+Sqrt[2])*(3+2*Sqrt[2])^n // Simplify
a1[n_] := ((3-2*Sqrt[2])^n+(3+2*Sqrt[2])^n)/2 // Simplify
Flatten[MapIndexed[{a0[#], a1[#]} &,Range[20]]] (* Gerry Martens, Jul 11 2015 *)

A041010=contfracpnqn(c=contfrac(sqrt(8)),#c)[1,][^-1] \\ Discard possibly incorrect last element. NB: a(n)=A041010[n+1]! For more terms use:
A041010(n)={n<#A041010|| A041010=extend(A041010, [-1,0,6,0]~, n\.8); A041010[n+1]}
extend(A,c,N)={for(n=#A+1,#A=Vec(A,N), A[n]=A[n-#c..n-1]*c);A} \\ (End)

a(n) = 6*a(n-2) - a(n-4).
a(2n) = a(2n-1) + a(2n-2), a(2n+1) = 4*a(2n) + a(2n-1).
a(2n) = A001333(2n), a(2n+1) = 2*A001333(2n+1).
G.f.: (2+3*x+2*x^2-x^3)/(1-6*x^2+x^4).
a(n) = A001333(n+1)*A000034(n+1). - R. J. Mathar, Jul 08 2009
From Gerry Martens, Jul 11 2015: (Start)
Interspersion of 2 sequences [a0(n),a1(n)] for n>0:
a0(n) = -((3-2*sqrt(2))^n*(1+sqrt(2))) + (-1+sqrt(2))*(3+2*sqrt(2))^n.
a1(n) = ((3-2*sqrt(2))^n + (3+2*sqrt(2))^n)/2. (End)

Entry improved by Michael Somos
Initial term 1 removed and b-file, program and formulas adapted by Georg Fischer, Jul 01 2019
Cross-references added by M. F. Hasler, Nov 02 2019

A265795 Denominators of primes-only best approximates (POBAs) to sqrt(8); see Comments.

Original entry on oeis.org

2, 2, 5, 7, 11, 59, 127, 163, 233, 653, 991, 1597, 11447, 12671, 70489

Offset: 1

Clark Kimberling, Dec 29 2015

Suppose that x > 0. A fraction p/q of primes is a primes-only best approximate (POBA), and we write "p/q in B(x)", if 0 < |x - p/q| < |x - u/v| for all primes u and v such that v < q, and also, |x - p/q| < |x - p'/q| for every prime p' except p. Note that for some choices of x, there are values of q for which there are two POBAs. In these cases, the greater is placed first; e.g., B(3) = (7/2, 5/2, 17/5, 13/5, 23/7, 19/7, ...). See A265759 for a guide to related sequences.

			The POBAs to sqrt(8) start with 7/2, 5/2, 13/5, 19/7, 31/11, 167/59, 359/127, 461/163, 659/233. For example, if p and q are primes and q > 59, then 167/59 is closer to sqrt(8) than p/q is.

Cf. A000040, A010466, A265759, A265790, A265791, A265792, A265793, A265794.

x = Sqrt[8]; z = 1000; p[k_] := p[k] = Prime[k];
t = Table[Max[Table[NextPrime[x*p[k], -1]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tL = Select[d, # > 0 &] (* lower POBA *)
t = Table[Min[Table[NextPrime[x*p[k]]/p[k], {k, 1, n}]], {n, 1, z}];
d = DeleteDuplicates[t]; tU = Select[d, # > 0 &] (* upper POBA *)
v = Sort[Union[tL, tU], Abs[#1 - x] > Abs[#2 - x] &];
b = Denominator[v]; s = Select[Range[Length[b]], b[[#]] == Min[Drop[b, # - 1]] &];
y = Table[v[[s[[n]]]], {n, 1, Length[s]}] (* POBA, A265794/A265795 *)
Numerator[tL]   (* A265790 *)
Denominator[tL] (* A265791 *)
Numerator[tU]   (* A265792 *)
Denominator[tU] (* A265793 *)
Numerator[y]    (* A265794 *)
Denominator[y]  (* A265795 *)

a(13)-a(15) from Robert Price, Apr 06 2019

A384214 Decimal expansion of the volume of a gyroelongated square cupola with unit edge.

Original entry on oeis.org

6, 2, 1, 0, 7, 6, 5, 7, 9, 2, 0, 3, 9, 2, 0, 0, 0, 3, 6, 6, 5, 8, 2, 2, 8, 8, 3, 3, 4, 5, 9, 8, 0, 7, 3, 1, 6, 9, 6, 0, 1, 0, 0, 3, 2, 0, 9, 1, 3, 7, 4, 5, 1, 7, 8, 3, 6, 4, 1, 8, 1, 7, 0, 5, 4, 3, 7, 9, 9, 6, 0, 4, 6, 7, 0, 8, 9, 3, 8, 4, 9, 5, 9, 9, 9, 4, 2, 7, 1, 3

Offset: 1

Paolo Xausa, May 23 2025

The gyroelongated square cupola is Johnson solid J_23.

			6.21076579203920003665822883345980731696010032091...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Gyroelongated square cupola.

Cf. A384215 (surface area).

Cf. A002193, A010466, A179587, A179638, A384142.

First[RealDigits[1 + Sqrt[8]/3 + 2/3*Sqrt[4 + Sqrt[8] + 2*Sqrt[146 + 103*Sqrt[2]]], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J23", "Volume"], 10, 100]]

Equals 1 + (2/3)*sqrt(2) + (2/3)*sqrt(4 + 2*sqrt(2) + 2*sqrt(146 + 103*sqrt(2))) = 1 + A010466/3 + (2/3)*sqrt(4 + A010466 + 2*sqrt(146 + 103*A002193)).

Equals the largest real root of 6561*x^8 - 52488*x^7 + 113724*x^6 - 9720*x^5 - 1616922*x^4 + 396360*x^3 + 1537020*x^2 - 178632*x - 3391.

A384215 Decimal expansion of the surface area of a gyroelongated square cupola with unit edge.

Original entry on oeis.org

1, 8, 4, 8, 8, 6, 8, 1, 1, 6, 2, 5, 9, 0, 5, 7, 6, 5, 6, 5, 2, 4, 0, 6, 0, 9, 1, 5, 5, 9, 4, 8, 7, 5, 7, 9, 9, 1, 8, 5, 3, 3, 7, 0, 0, 1, 9, 8, 0, 5, 7, 9, 9, 2, 8, 6, 6, 3, 2, 3, 9, 4, 3, 7, 3, 2, 4, 1, 1, 3, 0, 0, 4, 1, 4, 6, 8, 2, 1, 4, 2, 6, 3, 1, 0, 6, 5, 0, 6, 0

Offset: 2

Paolo Xausa, May 23 2025

The gyroelongated square cupola is Johnson solid J_23.

			18.4886811625905765652406091559487579918533700198...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Wikipedia, Gyroelongated square cupola.

Cf. A384214 (volume).

Cf. A002194, A010466.

First[RealDigits[7 + Sqrt[8] + Sqrt[75], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J23", "SurfaceArea"], 10, 100]]

Equals 7 + 2*sqrt(2) + 5*sqrt(3) = 7 + A010466 + 5*A002194.

Equals the largest root of x^4 - 28*x^3 + 128*x^2 + 952*x - 1244.

A385258 Decimal expansion of the volume of a gyroelongated square bicupola with unit edge.

Original entry on oeis.org

8, 1, 5, 3, 5, 7, 4, 8, 3, 3, 6, 2, 1, 2, 6, 3, 4, 0, 2, 5, 2, 6, 0, 2, 1, 3, 1, 6, 2, 6, 6, 2, 7, 2, 7, 0, 2, 6, 7, 3, 2, 1, 4, 9, 0, 4, 4, 9, 8, 3, 7, 7, 2, 2, 7, 1, 4, 8, 6, 3, 4, 8, 6, 4, 0, 9, 8, 4, 8, 4, 3, 6, 5, 6, 8, 3, 6, 7, 6, 5, 2, 1, 8, 9, 9, 6, 8, 5, 4, 9

Offset: 1

Paolo Xausa, Jun 26 2025

The gyroelongated square bicupola is Johnson solid J_45.

			8.1535748336212634025260213162662727026732149...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Gyroelongated square bicupola.

Cf. A385259 (surface area).

Cf. A002193, A010466, A179587, A384214, A384287, A385256.

First[RealDigits[2/3*(3 + 2*# + Sqrt[2*(2 + # + Sqrt[146 + 103*#])]) & [Sqrt[2]], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J45", "Volume"], 10, 100]]

Equals (2/3)*(3 + 2*sqrt(2) + sqrt(2*(2 + sqrt(2) + sqrt(146 + 103*sqrt(2))))) = (2/3)*(3 + A010466 + sqrt(2*(2 + A002193 + sqrt(146 + 103*A002193)))).

Equals the largest real root of 6561*x^8 - 104976*x^7 + 594864*x^6 - 1384128*x^5 - 552096*x^4 + 1569024*x^3 + 246528*x^2 - 119808*x + 4352.

A011006 Decimal expansion of 4th root of 8.

Original entry on oeis.org

1, 6, 8, 1, 7, 9, 2, 8, 3, 0, 5, 0, 7, 4, 2, 9, 0, 8, 6, 0, 6, 2, 2, 5, 0, 9, 5, 2, 4, 6, 6, 4, 2, 9, 7, 9, 0, 0, 8, 0, 0, 6, 8, 5, 2, 4, 7, 1, 3, 5, 6, 9, 0, 2, 1, 6, 2, 6, 4, 5, 2, 1, 7, 1, 9, 4, 9, 8, 4, 9, 5, 0, 9, 9, 0, 7, 8, 0, 4, 4, 7, 9, 6, 2, 8, 6, 4, 8, 0, 0, 8, 3, 9, 8, 5, 8, 5, 0, 7

Offset: 1

N. J. A. Sloane

			1.68179283...

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Index entries for algebraic numbers, degree 4.

RealDigits[N[8^(1/4), 100]][[1]] (* Vincenzo Librandi, Apr 04 2013 *)

sqrtn(8,4) \\ Charles R Greathouse IV, Apr 15 2014

Equals 2*A228497 =2^(3/4) = sqrt(A010466). - R. J. Mathar, Jan 15 2021

A280533 Decimal expansion of 14*sin(Pi/14).

Original entry on oeis.org

3, 1, 1, 5, 2, 9, 3, 0, 7, 5, 3, 8, 8, 4, 0, 1, 6, 6, 0, 0, 4, 4, 6, 3, 5, 9, 0, 2, 9, 5, 5, 1, 2, 6, 6, 3, 2, 5, 2, 8, 9, 7, 7, 9, 6, 2, 7, 0, 3, 6, 2, 9, 3, 7, 4, 3, 6, 7, 8, 1, 8, 2, 2, 2, 5, 6, 3, 8, 9, 7, 2, 4, 8, 3, 9, 9, 6, 6, 2, 4, 6, 7, 0, 4, 4, 1, 3, 4, 7, 3, 6, 5, 1, 3, 0, 2, 1, 3, 8, 8, 8, 8, 2, 4, 5

Offset: 1

Rick L. Shepherd, Jan 04 2017

Decimal expansion of the ratio of the perimeter of a regular 7-gon (heptagon) to its diameter (largest diagonal).

			3.115293075388401660044635902955126632528977962703629374367818222563897248...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for algebraic numbers, degree 3.

Cf. For other n-gons: A010466 (n=4), 10*A019827 (n=5, 10), A280585 (n=8), A280633 (n=9), A280725 (n=11), A280819 (n=12).

Cf. A232736.

RealDigits[14*Sin[Pi/14], 10, 129][[1]] (* G. C. Greubel, Sep 20 2022 *)

PARI
```
14*sin(Pi/14)
```

numerical_approx(14*sin(pi/14), digits=127) # G. C. Greubel, Sep 20 2022

Equals 14*A232736.

A335930 Decimal expansion of the arclength on y = sin(x) from (0,0) to (Pi,0).

Original entry on oeis.org

3, 8, 2, 0, 1, 9, 7, 7, 8, 9, 0, 2, 7, 7, 1, 2, 0, 1, 7, 9, 0, 4, 7, 6, 2, 0, 8, 2, 1, 7, 1, 4, 4, 3, 2, 9, 1, 9, 0, 9, 9, 6, 7, 6, 1, 4, 6, 4, 7, 2, 7, 4, 7, 2, 1, 0, 8, 0, 4, 9, 6, 6, 5, 6, 7, 4, 7, 1, 9, 5, 8, 0, 1, 2, 1, 4, 3, 2, 9, 9, 2, 1, 0, 6, 6, 1, 8, 1, 0, 8

Offset: 1

Clark Kimberling, Jul 01 2020

Also the arclength between consecutive points of intersection of y = sin(x) and y = cos(x).

			arclength = 3.8201977890277120179047620821714432919099676146...

Cf. A335928, A335929, A335931, A335932.

Cf. A000796, A010466, A053004, A062539, A105419, A256667, A257407.

r = NIntegrate[Sqrt[1 + Cos[t]^2], {t, 0, Pi}, WorkingPrecision -> 200]
RealDigits[r][[1]]
First[RealDigits[Sqrt[8]*EllipticE[1/2], 10, 100]] (* Paolo Xausa, Nov 14 2024 *)

From Paolo Xausa, Nov 14 2024: (Start)
Equals Pi/A062539 + A062539 = A053004 + A062539.
Equals A010466*A257407. (End)
Equals A105419/2 = 2*A256667. - Hugo Pfoertner, Nov 14 2024

A194381 Numbers m such that Sum_{k=1..m} (<1/2 + k*r> - ) < 0, where r=sqrt(8) and < > denotes fractional part.

Original entry on oeis.org

1, 2, 3, 7, 8, 9, 13, 14, 15, 19, 20, 21, 25, 26, 27, 31, 32, 33, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89

Offset: 1

Clark Kimberling, Aug 23 2011

nonn

See A194368.

Cf. A010466, A194368, A194382, A194383, A194384.

r = Sqrt[8]; c = 1/2;
x[n_] := Sum[FractionalPart[k*r], {k, 1, n}]
y[n_] := Sum[FractionalPart[c + k*r], {k, 1, n}]
t1 = Table[If[y[n] < x[n], 1, 0], {n, 1, 100}];
Flatten[Position[t1, 1]]   (* A194381 *)
t2 = Table[If[y[n] == x[n], 1, 0], {n, 1, 300}];
Flatten[Position[t2, 1]]   (* A194382 *)
%/2  (* A194383 *)
t3 = Table[If[y[n] > x[n], 1, 0], {n, 1, 600}];
Flatten[Position[t3, 1]]   (* A194384 *)

cp's OEIS Frontend

A378390 Decimal expansion of the surface area of a deltoidal icositetrahedron with unit shorter edge length.

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A041010 Numerators of continued fraction convergents to sqrt(8).

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A265795 Denominators of primes-only best approximates (POBAs) to sqrt(8); see Comments.

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Mathematica

Extensions

A384214 Decimal expansion of the volume of a gyroelongated square cupola with unit edge.

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A384215 Decimal expansion of the surface area of a gyroelongated square cupola with unit edge.

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Mathematica

Formula

A385258 Decimal expansion of the volume of a gyroelongated square bicupola with unit edge.

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Mathematica

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A011006 Decimal expansion of 4th root of 8.

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A280533 Decimal expansion of 14*sin(Pi/14).

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